Critical mass for an attraction–repulsion chemotaxis system

Guo, Qian; Jiang, Zhaoxin; Zheng, Sining

doi:10.1080/00036811.2017.1366989

Cited by 23 publications

(25 citation statements)

References 6 publications

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“…Conversely, the sole insertion in the classical Keller–Segel model of a repulsive effect coming from another chemical substance does not suffice to avoid δ‐formations for the cells' density. More precisely, confining our attention to the linear diffusion case

A(u,v,w)\equiv 1

, and fixing

B(u,v,w)=-\chi \nabla u

and

C(u,v,w)=\xi \nabla u

(

\chi , \xi >0

D(u,v,w)\equiv 0

and production rates

E(u,v,w)=\alpha u-\beta v

F(u,v,w)=\gamma u-\delta w

(

\alpha ,\beta ,\gamma ,\delta >0

), for

\tau =0

, we have that the sign of

\xi \gamma -\chi \alpha

(positive repulsion prevails over attraction, negative attraction prevails over repulsion) establishes whether system (1.1) has unbounded solutions or all solutions are bounded: see the significant contribution [21] and [5, 6, 15 22] for some details on the issue.…”

Section: Introduction and Presentation Of The Main Resultsmentioning

confidence: 99%

Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

Viglialoro

2021

Mathematische Nachrichten

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This paper is dedicated to the attraction‐repulsion chemotaxis‐system 65.0pt{ut=Δu−χ∇·false(u∇vfalse)+ξ∇·false(u∇wfalse)inΩ×(0,Tmax),0=Δv+f(u)−βvinΩ×(0,Tmax),0=Δw+g(u)−δwinΩ×(0,Tmax),\begin{equation} \hspace*{65pt}\left\{ \def\eqcellsep{&}\begin{array}{ll} u_{ t}=\Delta u -\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w) & \text{in }\Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta v+f(u)-\beta v & \text{in } \Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta w+g(u)-\delta w & \text{in } \Omega \times (0,T_{\mathrm{max}}), \end{array} \right. \end{equation}defined in Ω, a smooth and bounded domain of double-struckRn$\mathbb {R}^n$, with n≥2$n\ge 2$. Moreover, β,δ,χ,ξ>0$\beta ,\delta ,\chi ,\xi >0$ and f,g$f, g$ are suitably regular functions generalizing, for u≥0$u\ge 0$ and α, γ>0$\gamma >0$ the prototypes f(u)=αus$f(u)=\alpha u^s$, s>0$s>0$, and g(u)=γur$g(u)=\gamma u^r$, r≥1$r\ge 1$. We focus our analysis on the value Tmax∈(0,∞]$T_{\mathrm{max}}\in (0,\infty ]$, establishing the temporal interval of existence of solutions false(u,v,wfalse)$(u,v,w)$ to problem (⋄$\Diamond$). When zero‐flux boundary conditions are fixed, we prove the following results, all excluding chemotactic collapse scenarios under certain correlations between the attraction and repulsive effects describing the model. To be precise, for every α,β,γ,δ,χ>0$\alpha ,\beta ,\gamma ,\delta ,\chi >0$, and r>s≥1$r>s\ge 1$ (resp. s>r≥1$s>r\ge 1$), there exists ξ∗>0$\xi ^*>0$ (resp. ξ∗>0$\xi _*>0$) such that if ξ>ξ∗$\xi >\xi ^*$ (resp. ξ≥ξ∗$\xi \ge \xi _*$), any sufficiently regular initial datum u0(x)≥0$u_0(x)\ge 0$ (resp. u0(x)≥0$u_0(x)\ge 0$ enjoying some smallness assumptions) produces a unique classical solution false(u,v,wfalse)$(u,v,w)$ to problem (⋄$\Diamond$) which is global, i.e. Tmax=∞$T_{\mathrm{max}}=\infty$, and such that u, v and w are uniformly bounded. Conversely, the same conclusion holds true for every α,β,γ,δ,χ,ξ>0$\alpha ,\beta ,\gamma ,\delta ,\chi ,\xi >0$, 0

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A(u,v,w)\equiv 1

, and fixing

B(u,v,w)=-\chi \nabla u

and

C(u,v,w)=\xi \nabla u

(

\chi , \xi >0

D(u,v,w)\equiv 0

and production rates

E(u,v,w)=\alpha u-\beta v

F(u,v,w)=\gamma u-\delta w

(

\alpha ,\beta ,\gamma ,\delta >0

), for

\tau =0

, we have that the sign of

\xi \gamma -\chi \alpha

Section: Introduction and Presentation Of The Main Resultsmentioning

confidence: 99%

Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

Viglialoro

2021

Mathematische Nachrichten

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“…Wanting to provide some details in this frame, for the fully elliptic version (i.e τ = 0 in the equations for v and w) of model (1) these results are available in the literature. When linear growths of the chemoattractant and the chemorepellent are considered, h(u, v) = αu, α > 0, and k(u, v) = γu, γ > 0, the value ξγ − χα, measuring the difference between the repulsion and attraction contributions, is critical for n = 2: particularly, if ξγ − χα > 0 (repulsion dominated regime), in any dimension all solutions to the model are globally bounded, whereas for ξγ − χα < 0 (attraction dominated regime) unbounded solutions can be detected (see [4,10,17,18,25] for some connected studies). On the other hand, for more general production laws, respectively h and k generalizing the prototypes h(u, v) = αu s , s > 0, and k(u, v) = γu r , r ≥ 1, we are only aware of the following recent result, valid for n ≥ 2 ( [19]): for every α, β, γ, δ, χ > 0, and r > s ≥ 1 (resp.…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

Uniform in time $L^\infty$-estimates for an attraction-repulsion chemotaxis system with double saturation

Frassu¹,

Galván²,

Viglialoro³

2021

Preprint

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In this paper we focus on this attraction-repulsion chemotaxis model with consumed signalsformulated in a bounded and smooth domain Ω of R n , with n ≥ 2, for some positive real numbers χ, ξ and with Tmax ∈ (0, ∞].Once equipped with appropriately smooth initial distributions u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0 and w(x, 0) = w 0 (x) ≥ 0, as well as Neumann boundary conditions, we establish sufficient assumptions on its data yielding global and bounded classical solutions; these are functions u, v and w, with zero normal derivative on ∂Ω × (0, Tmax), pointwisely satisfying the equations in problem (✸) with Tmax = ∞. This is proved: 1) for any such initial data, whenever χ and ξ belong to bounded and open intervals, depending respectively on v 0 L ∞ (Ω) and w 0 L ∞ (Ω) ; 2) for a wider (but also bounded) interval of ξ, provided a further largeness assumption on the minimum of w 0 in Ω is imposed; 3) for an interval of ξ arbitrarily extendable, whenever the mentioned largeness assumption on the minimum of w 0 in Ω is preserved, and as long as also w 0 L ∞ (Ω) is allowed to indefinitely increase. Lastly, we conclude the article discussing our results in the frame of others available in the literature; specifically, in the more restricted class of initial data u 0 , v 0 , w 0 such that χv 0 − ξw 0 ≥ 0, model (✸) may be also faced by considering a natural transformation and then adapting already known theorems. Again with regard to the boundeness of its solutions, the aforementioned transformation yields such a property for sharper conditions on χv 0 − ξw 0 L ∞ (Ω) than those directly obtained with our approach, when the initial data are taken in accordance to items 1) and 2). Conversely, for those data complying with restrictions in item 3), the situation changes and, making the most from the repulsive coefficient ξ, our strategy provides a milder condition.

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“…As to the connection between the classical blow-up in the L ∞ (Ω)-norm of solutions to (1), i.e. relation (3), and that in the L 2 (Ω)-norm (and in general in the L p (Ω)-norm, p > 1), i.e. in the sense that lim sup Ω u 2 dx ր ∞ as t ց t * , we want to observe that once it is assumed that Ω is a bounded domain, we only know that u(·, t) L 2 (Ω) ≤ |Ω| 1 2 u(·, t) L ∞ (Ω) , so that if a solution blows up in the L 2 (Ω)-norm, it does in the L ∞ (Ω)-norm.…”

Section: Remarkmentioning

confidence: 99%

“…(See also [9].) • In the recent paper [3] it is established that for n = 2 and χα − ξγ > 0, the value 4π χα−ξγ is the critical mass for the attraction-repulsion chemotaxis system (1) through which it is possible to identify global boundedness or possible finite time blow-up of solutions.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Explicit lower bound of blow–up time for an attraction–repulsion chemotaxis system

Viglialoro

2019

Journal of Mathematical Analysis and Applications

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In this paper we study classical solutions to the zero-flux attractionrepulsion chemotaxis-systemwhere Ω is a smooth and bounded domain of R 2 , t * is the blow-up time and α, β, γ, δ, χ, ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable smallness assumptions on the initial data u(x, 0) = u 0 ∈ C 0 (Ω), system (♦) has a unique classical solution which becomes unbounded as t ր t * . The main result of this investigation is to provide an explicit lower bound for t * estimated in terms of Ω u 2 0 dx and attained by means of well-established techniques based on ordinary differential inequalities.2010 Mathematics Subject Classification. 35A01, 35B44, 35K55, 35Q92, 92C17.

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Critical mass for an attraction–repulsion chemotaxis system

Cited by 23 publications

References 6 publications

Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

Uniform in time $L^\infty$-estimates for an attraction-repulsion chemotaxis system with double saturation

Explicit lower bound of blow–up time for an attraction–repulsion chemotaxis system

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